If so then because p1*x^8 is going to be the greatest influence on the result for x > 1, and your p1 values in () cross zero, then you effectively cannot even meaningfully find the sign for f(x), which would suggest that you might have overfitted f(x) or might have attempted to use a polynomial model for something which cannot be approximated by a polynomial.

What is the range of x you are interested in? If it goes much beyond -1 to +1 then I don't think you are going to be able to find equations. In the range near -1 to +1 you just might be able to find an equation.

Not a chance. With those order 8 polynomial fits, by the time you reach x = 100, the K=5 curve has reached y = 10^14 and the K=20 curve has reached y = -10^13.

Calculate for a moment. At x=100, p1*x^8 is going to be p1*100^8 = p1*10^16 . In order for that to remain in the range 0 to 200, p1 must be non-negative and p1 can be at most roughly 200/(10^16), which is within ep of the two p1 values you show.

Either your fitting is giving you numeric nonsense or else the actual desired equation is very different from polynomial. Probably both.